Results: Plots and Tables

This report contains different plots and tables that may be relevant for analysing the results. Observe:

• The hypercube \(Y_N\) is placed in, is approx. \([0, 10000m]^p\), since for each subproblem \(Y_N^s \subset [0, 10000]^p\). That is, more subproblems create a larger hypercube for the master problem and the volume (number of integer points) grows exponentially in \(p\).
• Given \(|Y_N|\), we can find the hypercube as \[[l_1, u_1] \times \cdots \times [l_p, u_p],\] where \(l_i = \min_{y \in Y_N}(y_i)\) and \(u_i = \max_{y \in Y_N}(y_i)\).
• The number of integer points in the hypercube is \[\prod_{i = 1}^p (u_i - l_i + 1)\]

Statistics for alg1

Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\) (alg1).

The following instance/problem groups are generated given:

• \(p=2,\ldots, 5\). [4 options]
• \(m=2,\ldots 5\) where \(m\) is the number of subproblems. [4 options]
• All subproblems have the same method config or half have method u and l. [4 options]
• Five instances for each config. [5 options]

Status

• 976/1600 problems have been solved, i.e. 624 remaining.
• 970/976 problems have 5 instances solved for each configuration.
• 148/976 have not been classified.
• 81/976 have not been fully classified (only classified extreme).

Problems solved for the analysis

Note that the width of objective \(i = 1, \ldots p\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\). Check:

## # A tibble: 4 × 6
##       m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
##   <dbl>       <dbl>       <dbl>       <dbl>       <dbl>       <dbl>
## 1     2      19317.      19288.      19326.      19135.      18887.
## 2     3      28566.      28635.      28509.      28363.      27702 
## 3     4      38296.      38374.      38393.      37965.      37041.
## 4     5      47300.      47636.      47492.      46924.      44537.

Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l         61102.   170
## 2 m        483719.   315
## 3 u        222594.   315
## 4 ul        56146.   170

Does \(p\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     p mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     3248.    50
##  2 m          2     5580.    80
##  3 u          2     1091.    80
##  4 ul         2      946.    50
##  5 l          3    18814.    50
##  6 m          3   100415     80
##  7 u          3     9834.    80
##  8 ul         3     6358.    50
##  9 l          4    91921.    35
## 10 m          4   771608.    80
## 11 u          4   489860.    80
## 12 ul         4    90554.    35
## 13 l          5   173344.    35
## 14 m          5  1095511.    75
## 15 u          5   400722.    75
## 16 ul         5   171720.    35

Does \(m\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     m mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2    37198.   100
##  2 m          2    43615.   100
##  3 u          2    36622.   100
##  4 ul         2    35269.   100
##  5 l          3   148973.    40
##  6 m          3   367604.    80
##  7 u          3   304142.    80
##  8 ul         3   145011.    40
##  9 l          4    21163     20
## 10 m          4   935598.    80
## 11 u          4   371994.    80
## 12 ul         4     6950.    20
## 13 l          5    28539.    10
## 14 m          5   795527.    55
## 15 u          5   224798.    55
## 16 ul         5     7844.    10

Relative size of \(Y_N\)

Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.

Plots used in the paper